This site is the blogging component for my main site Crank Astronomy (formerly "Dealing with Creationism in Astronomy"). It will provide a more interactive component for discussion of the main site content. I will also use this blog to comment on work in progress for the main site, news events, and other pseudoscience-related issues.

Here we'll explore some more details about the two-body problem using Newton's laws and gravitation. An n-body code solves Newton's equations by computing the changing positions of multiple objects subjected to various forces. In this case, the forces used are the gravitational forces between the two bodies computed by Newton's Law of Gravitation:

Here M1 & M2 are the two masses and r12 is the distance between the. 'G' is the Newtonian gravitational constant. I provided movies of a simple 2-body problem in my first post on the n-body code (Doin' Real Science: Simulating Particles).

In this post, I'll examine the 2-body problem in more detail.

Setup

For these types of problems in science, we must first setup *Initial conditions* for the problem. The program will then compute the motions forward in time, updating velocities and positions based on the forces on the objects (in this case, gravity).

Let's start with two simple objects, which we call Primary and Secondary,

I'll specify the object speeds in astronomical units per year (AU/yr) and the masses relative to the mass of the Sun. To convert the speed to more familiar units:

1 au/yr = (149.598e6 km)/(365.25*24*3600 s) = 4.74 km/s

= (92.955e6 miles/(365.25*24 hr) = 10604 miles per hour

So to convert my speeds to more familiar values, multiply the AU/year value by 4.74 to get kilometers/second or 10604 to get miles/hour or 2.9455 to get miles per second.

In the computer program, I give the Primary and Secondary objects initial positions and velocities consistent with this setup. We call them INITIAL positions and velocities because they will change with time as the force acts on them. For simplicity, I place the Primary object at the origin (0, 0, 0), the *center* of our universe as we have defined it, and give it an initial velocity of zero (0, 0, 0).

I place the Secondary object at (1AU, 0, 0) and give it a velocity in the y-direction (perpendicular to the initial line between the Primary & Secondary, which defines the x-axis). For this case, I set the magnitude of the velocity to 60% of the speed of a circular orbit at the distance of 1AU.

To make the problem more general, I add an additional velocity of 1AU/yr in the y-direction for both of these objects. Here's a printout of the table of initial parameters:

The position vector is 'x' and the velocity vector is 'v'. The 'Q' column specifies that the planets have no significant electric charge.

Analytic Solution

Using Kepler's laws, we can compute a few specifics about our orbit before running the simulation. This will help provide a check on the accuracy of the simulation run. N-body problems accumulate small errors at each step of the computation so as they run for longer and longer times, their accuracy declines. Here's some Keplerian parameters based on the input values.

Because we chose a speed for the Secondary object that is less than the circular orbit velocity at 1AU, we see that our initial position will correspond to the apoapsis (or farthest distance) of our orbit from the Primary object. When we start running the simulation, we expect the Secondary object to fall *towards* the Primary object.

Numerical Solution

So we run the simulation in the coordinate system as we have defined. Here's a plot of the output dataset.

The red curve tracks the motion of the Primary and the green curve tracks the motion of the Secondary. The black dashed curve corresponds to the motion of the center of mass of the two bodies, computed at each time we've plotted.

We note several things about the motion

the primary object, while it started out at the origin with zero velocity, did not stay at the origin. This is a problem for Geocentrists since it will happen for all objects moving under mutual forces.

both masses move around the center of mass point, the CM lying on the line between them. These are the motions measured when searching for planets by the radial velocity method (wikipedia).

Remember that I gave both objects an initial motion of 1AU/yr in the y-direction. Yet the CM is moving slightly faster, 1.34 AU/yr. This is again due to the fact that gravitation is a mutual force and Newton's laws require an action & reaction.

But those unfamiliar with these types of problems, these paths just look like wavy lines. WHERE IS THE ORBIT?

Kepler's laws are actually defined in the reference frame were one of the objects is at rest, as I noted in the earlier post. So we must compute the RELATIVE positions of the objects. Here we do the case of the motion relative to the Primary (more massive) body.

x(1,ref), y(1,ref), z(1,ref) = position of object 1, in the coordinate system where the origin is object ref.

For the Primary body, this becomes:

x(1,ref) = x(1) - x(1) = 0.0
y(1,ref) = y(1) - y(1) = 0.0

So the Primary resides at the origin in this new coordinate system. Similarly, we compute the new position of the Secondary object at each time step:

With this plot, we can more easily compare the plot to the computed orbital period, semi-minor axis, and periapsis in the table above with the aid of the grid overlay. The secondary (green) completes the orbit between t=0.440 yr and t=0.441 yr, consistent with P=0.4407 yr computed from the input values. We also observe the speed-up of the Secondary as it approaches the Primary mass, as predicted by Kepler's Laws. I also plot the center-of-mass (black dashed line) in this reference frame. In this example, the CM moves in an ellipse around the primary mass.

Other Frames of Reference

Because mathematically, the frames are equivalent, we can also convert to the frame of the less massive object as well!

1. The orbital mechanics of Newton dictates the earth orbits the sun’s
center of mass in an ellipse, yet Newtonian mechanics states the earth
also orbits the solar system barycenter. As the solar system barycenter
is almost never at the center of mass of the sun, then the earth simply
cannot be orbiting the sun in an ellipse. Therefore Newton’s principle
of barycentric motion invalidates Kepler’s laws of elliptical motion.

But now we can see these motions are perfectly compatible. The only evidence Mr. Martin presents for his claim is his 'say so'.

In fact, I can also plot the motion in the original input frame (green & red) and overlay the motion of the secondary mass relative to the primary mass (magenta).

Notice that from the input frame, the Keplerian orbit is not fixed in space, but appears to be carried by the primary mass as the primary moves around the center of mass.

N-body codes are a regular tool for astronomy and celestial navigation. While I don't include many of the smaller forces (planetary oblateness, relativisitic effects, etc.) my tool is sufficiently accurate to demonstrate many principles in the application of Newton's laws with gravitation.

I've got a number of simulations of Geocentrists' claimed "problems" that I'm writing up for future posts.

But even this extended content did not cover all the items I would have liked to include, so I'll add a few notes, caveats, and qualifiers here.

Nature Rarely Accommodates the Pure Form
Think of how difficult is was for Isaac Newton to derive the pure form of F=ma, in a world surrounded by frictional forces. Similarly, one of the problems we run into is that phenomena such as the Pannekoek-Rosseland (P-R) field and the field of an offset rotating magnetic dipole are next to impossible to obtain in a 'pure' form in nature.

For the P-R field, any deviation from spherical symmetry will create a non-radial component to the electric field. This deviation from symmetry will start charges moving (a current) which will generate a magnetic field, which makes the motion even more complex. This complex set of motions may find a near-equilibrium configuration with the rest of the star's structure, perhaps initializing the star's magnetic dynamo and other plasma motions near the star's surface.

For the case of the rotating dipole, the configuration described in the podcast generates an electric field in a vacuum. But in reality, pulsars, stars, and even planets, are surrounded by a plasma. If an electric field is created by the dipole rotation in this environment, the charges in the plasma will start moving, distributing themselves in a configuration that tries to cancel the electric field. But the inertia of the particles will allow them to overshoot strict electrical neutrality, much the way it occurs in Langmuir's plasma oscillations (wikipedia). In configurations like pulsars, stars and planets, additional complexities are created by winds of charged particles, either streaming off the atmosphere of the pulsar or planet, or off a nearby star. The combination of all these complex process makes magnetospheric physics (wikipedia) one of the most complex fields of study in the space sciences.

Breaking Wind?
And the last qualifier I'll add is for the description of stellar winds driven by radiation pressure. While the bulk of the stellar wind is electrons and protons, they are virtually invisible to distant spectrographs as they have no well-defined spectral lines. Modern kinetic treatments of the solar wind include electric fields which can be created by regions of charged particle separation in the wind. But since the electric field is determined by the particle distribution, and it's easier to measure the particle distribution, the electric field is usually just a bookkeeping tool for tracking the forces in the flow.

In addition, elements heavier than helium in the stellar wind are more readily detected by remote sensing techniques since they have complex spectra. The energy-levels of these atoms interact with the photons streaming out from the star, which applies a net force on the atoms. For distant stars, we can detect these winds powered by the absorption and re-emission of the atoms.

Even More Goodies...
And there are some other processes which I did not have time to discuss in the podcast, but which are possible topics of future interest.

Ambipolar diffusion of solar wind plasma around structures, as well as photoelectric ionization on the Moon and asteroid surfaces can generate potentials of several hundred volts. These voltages can impact the safety of astronauts as well as satellites and other equipment. Evidence for Acceleration of Lunar Ions (1972)

The major point of the book is that modern physics has demonstrated that even what we think of as 'nothing', even empty space, is rippling with activity, virtual particle pairs popping in and out of existence. We have detected the effects of this process in laboratory experiments, and it provides a mechanism whereby a universe can form 'from nothing'.

Here I'll cover a few high-points of the book which are relevant to the topics of this blog.

In chapter 1, Krauss discusses Lemaitre's original proposal of an expanding universe (1927-1930) which were solutions of Einstein's field equations of General Relativity. He also talks about how in 1951 Pope Pius XII tried to use Lemaitre's work as evidence that scientists had proved the biblical account of Genesis (pg 5). Lemaitre, who had trained for the priesthood, opposed this effort by the Pope, recognizing that scientific theories are subjected to continual testing and that the expanding universe could be overturned by evidence that would come later. It should be noted that many Electric Universe supporters use Lemaitre's religious training as evidence that the Big Bang is inherently religious - but that's like arguing that Kepler's Laws are invalid as science since Kepler (Wikipedia) was a practicing astrologer.

Krauss also provided a nice demonstration of how ANY system that expands where the distance between particles grows at a rate proportional to that distance (also called homologous expansion) will have any location appear to be the 'center' of the expansion (pg 11-14). Various versions of this are around the web and I'll have a version for a future post, linking it to claims of the Biblical Geocentrists that galaxy surveys show that we are at the 'center' of the Universe.

The next few chapters goes into detail about the historical development of our cosmological knowledge and its connection to the sub-atomic world. Krauss covers the discovery of gravitational lensing to antiparticles and their connection to how we now understand that even space we would consider empty is bubbling with activity - where particles and anti-particles pop in-and-out of existence over incredibly short amounts of time (Wikipedia: vacuum energy). I have also described some of these connections in "The Cosmos In Your Pocket" and other posts in this blog.

One nice thing about a book by someone like Krauss, who is actually connected with the cosmology and particle physics community as a professional, is that it provides some insights into some of the 'professional' challenges created by the participants. In the 1990s, cosmological problems inspired some cosmologists to re-examine Einstein's cosmological constant (one of my professors in grad school actually had this as an exercise for the cosmology class they were teaching at that time). The cosmological constant implied that even the vacuum had energy, something which physicists understood from quantum mechanics, but cosmologists opposed on more philosophical grounds. Krauss relates how Saul Perlmutter claimed that he would prove space doesn't have energy (pg 80). Perlmutter's group and another group both set out to prove that Einstein's cosmological term was zero, and wound up demonstrating the exact opposite. The groups recently won the Nobel Prize for this effort (Nobel site).

Chapter six explores one of the energy problems in cosmology, pointing out how if we compute the Newtonian energy of all the galaxies, the energy of their motion combined with the potential energy of the cosmological gravitational field, the total energy adds up to precisely zero. This also appears to be a condition for the universe to be flat on a cosmological scale (pg 103). Krauss implies that these two conditions may be physically equivalent, but I've had discussions with some cosmologists who have heard this equivalence expressed, but are not sure if it is mathematically rigorous.

By Chapter seven, Krauss is examining the evolution of the Universe far into the future, when galaxies will have redshifted to undetectability. Astronomers in that time will have limited observational evidence for determining the expansion nature of the universe and might reach some incorrect conclusions about the nature of the cosmos. Krauss points out that these incorrect conclusions, while based on the limited evidence available, are not the same as the demonstrably false picture advocated by young-earth creationists (pg 118).

In Chapter 10, Krauss talks about the problem with our definition of energy on cosmological scales (pg 166). I've had issues with some cosmologists who word this poorly, claiming that the cosmological expansion violates conservation of energy. These types of statements are picked up by various crank science supporters as evidence against mainstream cosmological models (and always in favor of their cosmological model). In reality, the issue is far more subtle. General relativity does indeed define conserved quantities, and one of them has the mathematical form of energy when we define it on small scales (such as galaxies, etc.). However this mathematical form does not map well to cosmological scales. I've had some discussions/arguments with cosmologists about the best analogy for this, but we've yet to find one.

Theological Implications
Throughout the book, Krauss makes the point that these physical laws eliminate the need for a God to be the creator of the universe. In actual fact, I would say it only eliminates the need for an interventionalist God that actively intervenes in physical laws. It does not rule out a non-intervening God, where the true spiritual test is the ability of intelligent beings to deal with the physical limitations in the real universe as a test of character (what I like to call the God of the Kobayashi Maru Scenario, wikipedia, Memory Alpha). Some of this theology is described in Martin Gardner's “The Why's of a Philosophical Scrivener”, pp 263-264.

While written for a very general audience interested in a quick review of modern cosmology, and how we got there, I found the book an enjoyable read.
At the book signing, I asked Krauss about a possible invitation from Geocentrists for some future conference they were planning. His response was that he didn't think he had received such an invite, but that the Geocentrists would have to make it *really* worth his while to participate in one of their conferences.

Sunday, February 5, 2012

Anyone who's flown on a plane knows that once the aircraft reaching its cruising speed (roughly constant) and altitude, for the most part, you can walk around the cabin just as you would if you were on the ground. Your leg muscles apply force in the same way as if you were walking on the ground. You need to do nothing special to account for the fact that you are moving 600+ miles per hour relative to the Earth. If the aircraft is accelerating, changing altitude or speed, whether by turbulence or intent, you must expend more effort, but accelerated frames are a topic for another time.

Similarly, any child knows that if the car (or van) is traveling at a constant speed on straight and level highway, you can toss your ball up from your lap and it will fall back to your lap, same as if you were sitting in a chair at home, even though the van you are in might be moving 50+ miles per hour relative to the Earth. You get a similar experience traveling on a train or subway.

The reason for this is that Newton's Laws are the same in inertial reference frames. Inertial frames are defined as non-accelerating, or moving at a constant velocity. A consequence of this fact is that Newton's Laws DO NOT DISTINGUISH ANY PREFERRED INERTIAL FRAME.

where 'F' is the applied force, 'm' is the mass of the object to which the force is being applied, and 'a' is the acceleration, or rate of change of velocity, 'v' (velocity is the rate of change of position). Mathematically, these are written using derivative (wikipedia) notation:

a = dv/dt = d^2x/dt^2

and the acceleration is the second derivative of position, x, with respect to time.

If the force, 'F', and mass, 'm', are constants, the most general solutions to this equation is, for some future time, t>0, for velocity, v:

v = dx/dt = B + (F/m)*t

and for position, x:

x = A + B*t + (1/2)*(F/m)*t^2

where 'A' & 'B' are called 'arbitrary constants' in the mathematical sense. In practice, these constants are determined from initial conditions (wikipedia) of the system under consideration. Beyond that, these constants are truly arbitrary - you can choose them with any origin, or zero point, that is convenient for your problem - the corner of your house, the center of the city, the center of the Earth, the center of Mars, or even the center of the Galaxy, or the center of a galaxy a million light-years away.

Note that the variable 'B' represents a velocity, so in addition to Newton's Laws being the same regardless of your POSITION in space, the laws also do not require a favored VELOCITY in space.
Note that if F = 0, the equation reduces to the spatial component of the Galilean transformation (wikipedia):

x = A + B*t

This equation solves the problem in one spatial dimension, but you can expand the case to three-dimensions by writing similar equations for the y and z directions, which are perpendicular to the x-direction, and each other.

Engineering Implications
Newton's Laws, and their invariant properties under coordinate transformations, have been experimentally tested for over three hundred years and have huge practical implications. They get tested with the construction of probably every mechanical device.

The most important of these practical applications is that in inertial frames, we can build devices that will work exactly the same if they are moved (accelerated) to another inertial frame. We can test a rover robot on the surface of the Earth, then transport it to Mars, and the forces required for it to move are fundamentally the same (adjusting for the different surface gravity, ground texture, etc.) - the torque driving the wheels moving it forward does not need to consider that the rover is on a planet moving many kilometers per second relative to the Earth. Similarly, thrusters adjusting the course of spacecraft moving very fast in the distant solar system (wikipedia: New Horizons) impart the same accelerations to the satellite as if it were in Earth orbit.

If the Earth were physically preferred reference frame, as claimed by Geocentrists, we would expect these principles to function differently when moving relative to the Earth, or at great distance from the Earth. The fact that these devices function according to the same physical laws we've discovered on the Earth is evidence that the Earth is NOT a physically preferred reference frame. A scientist on Mars will derive the exact same physical laws as a scientist on the Earth.

If Geocentrists want to claim that certain devices work in these other remote locations because we've designed them to work that way, the statement carries with it the implied assumption that somehow human technology violates the laws of physics. That is utter nonsense. The technological progress human society has enjoyed over the past three hundred years is an outgrowth of our ability to understand those physical laws and work within their constraints.

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About Me

I obtained my doctorate in physics and astronomy in 1994. I currently work in scientific data visualization for the media and public outreach. For more information on how I became involved in the creationism issue, visit my main page